We will present a series of
postulates pertinent to what was initially called wave mechanics. This is the
formulation that followed from Schrodinger equation in 1926.

However there are other
formulations of QM with varying degrees of abstraction. One of them was
formulated by Werner Heisenberg , Pascual Jordan and Max Born, called matrix mechanics, one or
two years before Schrodinger published his equation.

Eventually we will also
touch upon matrix mechanics.

Postulate A. The state of a system is represented by a wave function Ψ (q1 , q2…
qn ,t) , here qi can be for example the independent
cartesian coordinates of a system. Ψ may be a complex function. And it may also
may have a “spinor” part when account is taken of the spin property of
electrons. The spinor part is not a function of the coordinates qi .
It represents an internal degree of freedom like for example the charge or mass
of a particle.

A probability density is
defined by Ψ *, where Ψ * is the complex conjugate.

dP = Ψ * Ψ dq1
dq2 … dqn = Ψ * Ψ dτ ( dτ is the volume
element) is the probability that a measurement of the coordinates will give
values lying in the range (q1 , q1 +dq1 )…. (qn
, qn +dqn) . The probability is usually normalized such
that

∫ dP = ∫ Ψ * Ψ
dτ = 1.
(1)

Examples:

a) plane wave

A plane wave of the form
Ψ(x,t) = C * exp[ i( kx-ωt) ] can represent partially a beam of free particles with a well
defined linear momentum p traveling to the right , i.e + X axis. Here k = 2π/λ =
2π p/h and ω = v k . The expression for Ψ gives no details as to
the spread or cross section of the beam in the Y-Z plane. So this is in an
idealization. The constant C is determined by the normalization

∫ Ψ* Ψ dx = C* C (xf-xi) =N particles . If (xf-xi )
= L , the length of the space over which the particles manifest
their wave motion , then C2 = N/L and C= (N/L)1/2
. This option is called box normalization. There is another one ,
involving Dirac's delta function , but we won't go into that at this moment.

The normalized plane wave would be

Ψ (x,t) = (N/L)1/2
exp[ i(kx-ωt) ] .
(2)

Since Ψ2 is
constant , the particles have a uniform probability of being anywhere along the
X axis.

Setting N=1 and
h'k= p , E= h'ω in (2) the wave function takes the form

Ψ (x,t) = (1/L)1/2
exp[ (i/h') (px-Et) ]
. (3)

The result of taking the
following derivatives of Ψ , -ih'∂ /∂x ,
i h'∂ /∂t , justifies calling them the momentum operator and the energy operator
respectively.

-ih'∂ Ψ/∂x = p Ψ
, (4)

i h'∂ Ψ/∂t = E Ψ
. (5)

Later we will use the notion
of

p (variable) → poperator = -ih'∂ /∂x ,
(6)

and

E(variable) → E operator = i h'∂ /
∂t
. (7)

The energy operator is called
the hamiltonian and is denoted with the letter H.

The use of these two
operators will be the guiding principle in constructing Schrodinger equation.
In this very special case

Ψ is an eigenfunction of the
linear momentum and of the energy. It means that

popΨ = p Ψ and H Ψ = E Ψ .
. (8)

b) particle in the box

A particle confined in a
one dimensional box of length L has an un-normalized wave function

Eq (43) is symmetric in α and β.
Consider for example β fixed and α variable. Equation (43) can not have real
roots and therefore the discriminant of the quadratic must be less than or equal
to zero . This means that,

h'2 β2 - 4 < x2 > < p2 > β2
≤ 0 ,

< x2 > < p2 > ≥ (h'/2)2 .
(44)

Taking the square root and labeling
∆x ≡ < x2 >1/2 , ∆x≡ < p2 >1/2
. The notation ∆x and ∆p do not refer to a single measurement but to
a statistical measurement. With this notation the indeterminacy principle is
written as ,

∆x ∆p ≥ (h'/2)
.
(45)

Examples. Taken from reference 1.

Problem 1. Find the coordinate and p
distribution corresponding to the wave function

Ψ(x) = (α/π)1/4 exp( i k0 x) exp[-(α/2) (x-x0)2
] , (46-a)

or Ψ(x) = (α/π)1/4 exp( i
p0 x/h') exp[-(α/2) (x-x0)2
] , (46-b)

where h'k0 = p0 and
α ~ 1/L2 . The coordinate wipes out the phase part , i.e. the
exponential exp(ik0 x)

Equations (53) and (54)
state the obvious fact that the total energy in this case is the
kinetic energy.

The time dependent
Schrodinger equation is the necessary tool in scattering
problems and time dependent perturbations.This is very much the
situation in the quantum theory of radiation where the
electromagnetic potentials may be functions of time i.e. Φ(t)
and A(t).

Eq (52) has the
form of an initial value problem , for if Ψ(x,t=0)= Ψ
(x) (italics have been used) and if H Ψ (x)
= EΨ (x)
then

is Schrodinger's time independent equation.
The hamiltonian operator is H = -(h'2 /(2m) )∂2
/ ∂x2 + U(x). In general this is a boundary
value problem. Both Ψ
(x)and E are unknowns in
stationary problems where the time parameter does not enter. Conditions have to be imposed on the allowed
solutions Ψ
(x) resulting in general in a set Ψn (x)
and the corresponding quantized energies En
, n is the quantum number. The allowed values of n emanate from
the specific boundary conditions and are not imposed a priori.

Example: We give a numerical integration
example of i h' ∂ Ψ (x,t)/∂t = H Ψ(x,t)
using the algorithm (55).

The following fact is unknown to us, at
this stage of the Lectures . The function Ψ(x,t) =N exp(-x2 /2) exp(-iEt/h')
is the time dependent solution for the harmonic
oscillator ground state. N is a normalization constant obtained from
requiring

One can also find E from the time evolution of the numerical
solution Ψ (x,t) = g(x) cos(Et/h')
+ i g(x) sin(Et/h') .

Using the real part of the solution one gets , (with h'=1)
cos(Et) = real part Ψ (x,t) /g(x)
and

E=(1/t) arc cos{ real part Ψ (x,t) /g(x)
}. A numerical example is provided below.Of course not all estimates of E are
the same and as the numerical solution becomes poor especially at the
extremes so does the value of E.

Sage code

psi=exp(-x^2/2) ; v=(1/2)*x^2;
-(1/2)*diff(psi,x,2)+v*psi

1/2*e^(-1/2*x^2)

Fig 3-9. Time dependent solution of the harmonic oscillator
ground state, at t=0.

This a continuity equation analogous to
the one obtained in electromagnetism for the charge and current densities. Lets
generalize to three dimensions. The probability density is ρ = Ψ Ψ*
~ 1/L3 , and define the probability current density by

Two states systems provide, what in a
sense may be, the simplest examples of the superposition principle.

Here a state vector ,
│Ψ > is represented by the superposition of just two basis states, which are
ortho normal,

call them │1> and │2> ,

│Ψ > = c1│1 > + c2│2 > .
(66)

This notation is called bracket or
Dirac's notation. │Ψ > (the ket) represents a column vector while < Ψ│, (the
"bra") represents a row formed by the complex conjugate of │Ψ > .
The coefficients are given by , c1 = < 1│Ψ > , c2 = < 2│Ψ
>.

At this stage Schrodinger's equation
has not been invoked and no constant h' is used . One does know for a
physical fact, that certain systems can show two states or a mixture of
them. Such is the case of light with two planes of polarization X or Y or
right /left polarization. The electron spin also shows two projections
along the magnetic field called spin up or down.

Fig 3-13. A light wave propagating in the Z direction with the
electric field polarized θ degrees
with respect to X.

The situation shown in Fig 3-13 can be
represented with the abstract vectors,

The normalization of < Ψ│Ψ > = k2
E2 cos 2(θ ) + k2 E2 sin 2(θ
) =1 , requires that E2 /(8π) = h'ω/V . This states
that the energy density of the electromagnetic field (CGS units) equals
the energy of the photon ( h'ω ) divided by the volume V. It also ties a classical expression
,energy density of the field, to the quantum concept of the photon.